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<meta name="description" content="第二章 线性表线性表：表内数据类型相同，有限序列 本章将以总结的形式展现： 2.1 顺序表与链式表的区别     顺序表 链式表     存取 随机存取 顺序存取   结构 顺序存储（连续） 随机存储（不连续）   空间分配 静态存储（可以动态分配） 动态存储   操作 查找 O(1) ,插入和删除O（n） 查找 O(n) ,插入和删除O（1）   缺点 插入删除不便，长度不可以改变 查找速度慢，">
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<meta property="og:description" content="第二章 线性表线性表：表内数据类型相同，有限序列 本章将以总结的形式展现： 2.1 顺序表与链式表的区别     顺序表 链式表     存取 随机存取 顺序存取   结构 顺序存储（连续） 随机存储（不连续）   空间分配 静态存储（可以动态分配） 动态存储   操作 查找 O(1) ,插入和删除O（n） 查找 O(n) ,插入和删除O（1）   缺点 插入删除不便，长度不可以改变 查找速度慢，">
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        <ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link" href="#%E5%9B%9E%E5%BD%92"><span class="toc-text">回归</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E5%AE%9A%E4%B9%89"><span class="toc-text">定义</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92"><span class="toc-text">线性回归</span></a><ol class="toc-child"><li class="toc-item toc-level-5"><a class="toc-link" href="#1-%E6%9E%84%E9%80%A0%E6%A8%A1%E5%9E%8B%E5%87%BD%E6%95%B0"><span class="toc-text">1.构造模型函数</span></a></li><li class="toc-item toc-level-5"><a class="toc-link" href="#2-%E6%9E%84%E9%80%A0%E6%8D%9F%E5%A4%B1%E5%87%BD%E6%95%B0"><span class="toc-text">2.构造损失函数</span></a></li><li class="toc-item toc-level-5"><a class="toc-link" href="#3-%E6%8D%9F%E5%A4%B1%E5%87%BD%E6%95%B0%E6%9C%80%E5%B0%8F%E5%8C%96"><span class="toc-text">3.损失函数最小化</span></a></li></ol></li></ol></li></ol></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E5%AE%9E%E7%8E%B0%E5%AE%9E%E4%BE%8B%E4%BB%A3%E7%A0%81"><span class="toc-text">实现实例代码</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E5%B0%8F%E7%BB%93"><span class="toc-text">小结</span></a></li></ol>
    
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        <h1 id="回归"><a href="#回归" class="headerlink" title="回归"></a>回归</h1><p>​     </p>
<p><img src="https://moluggg.oss-cn-qingdao.aliyuncs.com/img/360截图20190821191648291.jpg" alt=""></p>
<p>如图，正方形斜线表示一个函数处理的黑盒，将输入的局进行处理，得到输出数据。</p>
<p>那回归是什么意思呢？其实说白了，就是这个黑盒子输出的结果是个连续的值。如果输出不是个连续值而是个离散值那就叫分类。</p>
<p>注意：一定要区别分类和回归的区别</p>
<p>Example：</p>
<p>比如我告诉你我这里有间房子，这间房子有40平，在地铁口，然后你来猜一猜我的房子总共值多少钱？这就是连续值，因为房子可能值80万，也可能值80.2万，也可能值80.111万。再比如，我告诉你我有间房子，120平，在地铁口，总共值180万，然后你来猜猜我这间房子会有几个卧室？那这就是离散值了。因为卧室的个数只可能是1， 2， 3，4，充其量到5个封顶了，而且卧室个数也不可能是什么1.1， 2.9个。</p>
<h1 id="定义"><a href="#定义" class="headerlink" title="定义"></a>定义</h1><p>百度百科上的定义：（可以不看）回归分析是一种数学模型。当因变量和自变量为线性关系时，它是一种特殊的线性模型。 [1] </p>
<p>最简单的情形是一元线性回归，由大体上有线性关系的一个自变量和一个因变量组成；模型是Y=a+bX+ε（X是自变量，Y是因变量，ε是<a target="_blank" rel="noopener" href="https://baike.baidu.com/item/%E9%9A%8F%E6%9C%BA%E8%AF%AF%E5%B7%AE">随机误差</a>）。 [2] </p>
<p>通常假定随机误差的<a target="_blank" rel="noopener" href="https://baike.baidu.com/item/%E5%9D%87%E5%80%BC">均值</a>为0，<a target="_blank" rel="noopener" href="https://baike.baidu.com/item/%E6%96%B9%E5%B7%AE">方差</a>为σ^2（σ^2﹥0，σ^2与X的值无关）。若进一步假定随机误差遵从<a target="_blank" rel="noopener" href="https://baike.baidu.com/item/%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83">正态分布</a>，就叫做正态线性模型。一般的，若有k个自变量和1个因变量，则因变量的值分为两部分：一部分由自变量影响，即表示为它的函数，函数形式已知且含有未知参数；另一部分由其他的未考虑因素和随机性影响，即随机误差。</p>
<p>当函数为参数未知的线性函数时，称为线性回归分析模型；当函数为参数未知的非线性函数时，称为非线性回归分析模型。当自变量个数大于1时称为多元回归，当因变量个数大于1时称为多重回归。</p>
<h1 id="线性回归"><a href="#线性回归" class="headerlink" title="线性回归"></a>线性回归</h1><h5 id="1-构造模型函数"><a href="#1-构造模型函数" class="headerlink" title="1.构造模型函数"></a>1.构造模型函数</h5><p>​    线性回归的目的是，通过构造一个函数模型，让数据集到模型的距离尽可能的小，谓词，我们可以构造一个损失函数，表示为：</p>
<p><img src="https://moluggg.oss-cn-qingdao.aliyuncs.com/img/20190824155412.png" alt=""></p>
<p>形象理解即每个点到模型的距离的平方</p>
<p><img src="https://moluggg.oss-cn-qingdao.aliyuncs.com/img/1566386823967.png" alt="">)</p>
<p>​    在高中，我们一般设y=ax+b，且计算a，b的公式都已经给出，因此直接套公式就可以得出。而由于现实中影响的因素不只是一个，所以要综合考虑：</p>
<p><img src="/../../img/1566386869688.png" alt="1566386869688"></p>
<p>​    那如果用房价这个例子来解释这个式子的话，就更好理解了。假如X1代表的是房子的总面积，X2代表的是房子的房间数量，X3代表的是楼间距，X4代表的是离学校的距离，等等等等。那这个时候里面的θ1就代表房子总面积对房价的重要程度，那如果θ1是个比较大的数，那就是说明房子总面积越大，那房价可能就越高。如果θ1等于0，就说明房价的高低跟房子的总面积没有半毛钱关系。（其他的θ请自行脑补），简单表示由下：</p>
<p><img src="/../../img/1566386878608.png" alt="1566386878608"></p>
<p>其实损失函数也可以向量化，因为可以把所有的yi排成一列撸成个列向量叫y，y^i也是个列向量叫y ^。然后假设y-y ^这个列向量叫U。那其实y-y ^的平方和就是U的平方和。那U的平方和无非可以想象成U的转置和U做个矩阵乘法。所以进化后就是酱紫：<img src="/../../img/1566386890770.png" alt="1566386890770"></p>
<h5 id="2-构造损失函数"><a href="#2-构造损失函数" class="headerlink" title="2.构造损失函数"></a>2.构造损失函数</h5><p>构造模型函数之后，我们要计算损失函数，并使这个损失函数最小。</p>
<p><img src="/../../img/1566386903635.png" alt="1566386903635"></p>
<h5 id="3-损失函数最小化"><a href="#3-损失函数最小化" class="headerlink" title="3.损失函数最小化"></a>3.损失函数最小化</h5><p>我们常用的有两种方法来求损失函数最小化时候的θθ参数：一种是梯度下降法，一种是最小二乘法。</p>
<p><strong>梯度下降法</strong>，我们以后会讲。</p>
<p><a target="_blank" rel="noopener" href="https://blog.csdn.net/qq_32864683/article/details/80368135">最小二乘法</a>：构造损失函数，对函数求偏导，让值为0</p>
<p>构造损失函数后，为了使损失函数最小，常用的方法是求偏导，使导数为0，为什么不跟梯度下降那样麻烦呢？</p>
<p>梯度下降大多数解决的是非线性的问题，非线性的模型就没有确定的方向了。</p>
<h1 id="实现实例代码"><a href="#实现实例代码" class="headerlink" title="实现实例代码"></a>实现实例代码</h1><p>既然弄清了原理，那么实现就会显得非常简单；考虑到这个算法过程中使用矩阵乘法的次数很多，所以我使用了python语言以及调用numpy库来实现线性回归的算法，这里使用了<strong>sklearn库中的波士顿房价数据集</strong>，代码如下：</p>
<pre><code class="lang-python"># coding=utf-8
import numpy as np
import copy
from sklearn.datasets import load_boston#导入博士顿房价数据集
class LinerRegression:
    M_x = []  #
    M_y = []  #
    M_theta = []  # 参数向量
    trained = False
def __init__(self):
    pass

def regression(self, data, target):
    self.M_x = np.mat(data)
    # 每个向量添加一个分量1，用来对应系数θ0
    fenliang = np.ones((len(data), 1))
    self.M_x = np.hstack((self.M_x, fenliang))
    self.M_y = np.mat(target)
    M_x_T = self.M_x.T  # 计算X矩阵的转置矩阵
    self.M_theta = (M_x_T * self.M_x).I * M_x_T * self.M_y.T# 通过最小二乘法计算出参数向量
    self.trained = True

def predict(self, vec):
    if not self.trained:
        print(&quot;You haven&#39;t finished the regression!&quot;)
        return
    M_vec = np.mat(vec)
    fenliang = np.ones((len(vec), 1))
    M_vec = np.hstack((M_vec, fenliang))
    estimate = np.matmul(M_vec,self.M_theta)
    return estimate
if __name__ == &#39;__main__&#39;:
    # 从sklearn的数据集中获取相关向量数据集data和房价数据集target
    data, target = load_boston(return_X_y=True)
    lr = LinerRegression()
    lr.regression(data, target)
    # 提取一批样例观察一下拟合效果
    test = data[::51]
    M_test = np.mat(test)
    real = target[::51]#实际数值real
    estimate=np.array(lr.predict(M_test))#回归预测数值estimate
    #打印结果
    for i in range(len(test)):
        print(&quot;实际值:&quot;,real[i],&quot; 估计值:&quot;,estimate[i,0])
</code></pre>
<p>运行结果如下：</p>
<blockquote>
<p>实际值: 24.0  估计值: 30.00384337701234<br>实际值: 20.5  估计值: 23.972222848686908<br>实际值: 18.6  估计值: 19.79013683546172<br>实际值: 19.4  估计值: 17.28601893611827<br>实际值: 50.0  估计值: 43.189498436968506<br>实际值: 20.9  估计值: 21.69580886553699<br>实际值: 33.4  估计值: 35.56226856966656<br>实际值: 21.7  估计值: 22.7180660747829<br>实际值: 17.2  估计值: 13.707563692104959 </p>
</blockquote>
<h1 id="小结"><a href="#小结" class="headerlink" title="小结"></a>小结</h1><p>1.注意区别回归和分类</p>
<p>2.回归的算法步骤</p>
<p>3.认识损失函数</p>
<p>通俗理解线性回归：<a target="_blank" rel="noopener" href="https://blog.csdn.net/alw_123/article/details/82825785">https://blog.csdn.net/alw_123/article/details/82825785</a></p>

      
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